Trigonometry Examples

$y = \cos (\frac{π}{4} - x)$

Step 1

Use the form

$a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = - 1$

$c = - \frac{π}{4}$

$d = 0$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$1$

Step 3

Find the period of

$\cos (\frac{π}{4} - x)$ .

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Step 3.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace

$b$ with

$- 1$ in the formula for period.

$\frac{2 π}{| - 1 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between

$- 1$ and

$0$ is

$1$ .

$\frac{2 π}{1}$

Step 3.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 4

Find the phase shift using the formula

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

$\frac{c}{b}$ .

Phase Shift:

$\frac{c}{b}$

Step 4.2

Replace the values of

$c$ and

$b$ in the equation for phase shift.

Phase Shift:

$\frac{- \frac{π}{4}}{- 1}$

Step 4.3

Dividing two negative values results in a positive value.

Phase Shift:

$\frac{\frac{π}{4}}{1}$

Step 4.4

Divide

$\frac{π}{4}$ by

$1$ .

Phase Shift:

$\frac{π}{4}$

Phase Shift:

$\frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$2 π$

Phase Shift:

$\frac{π}{4}$ (

$\frac{π}{4}$ to the right)

Vertical Shift: None

Step 6